Calculus of variations and optimal control by A. A. Milyutin and N. P. Osmolovskii

By A. A. Milyutin and N. P. Osmolovskii

The speculation of a Pontryagin minimal is built for difficulties within the calculus of diversifications. the appliance of the concept of a Pontryagin minimal to the calculus of diversifications is a particular characteristic of this publication. a brand new concept of quadratic stipulations for a Pontryagin minimal, which covers damaged extremals, is constructed, and corresponding adequate stipulations for a powerful minimal are received. a few classical theorems of the calculus of adaptations are generalized

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This solution is strong for initial data f~ E LMJRvN x (JRvN\WN)) C Ll(JRvN x (JRvN\W N )) and weak (generalized) for arbitrary initial data from Ll (lR vN x (JR vN \ W N))' Proof. 7). 3 that for ffJv E LMJRvN x (lR vN \ W N)} this solution is strong. 6). For this purpose we introduce a functional (cp,f,v(t)) = J dxcp(x)fN(t, x), where cp(x) is continuously differentiable functions with compact support, equal to zero in a certain c-neighbourhood of the set WN. As cp is bounded and fN(t) is summable, this functional exists.

Xnl ) 44 CHAPTER I is finite. For F(t) E L~ we have (N) < 00. 18) can be used also for the description of systems of infinitely many particles (see Chapter III). 1 On the solutions of the steady BBGKY hierarchy As was already mentioned in the introduction to this chapter, all possible states of statistical systems can be classified as equilibrium or non-equilibrium. Equilibrium states are described by distribution functions that are special solutions of the steady BBGKY hierarchy. 19}). 1) was examined in a series of papers [GS].

2') for the Poisson bracket. Remark 6. The operator 1iN, as infinitesimal generator of the group SN(t) of isometric operators, is closed on V(1iN)' It acts differently at t > 0 and t < 0 and we have already proved that on the set Lli(JRvN x (JRvN\ W N)) it is given by the Poisson bracket with boundary conditions on 8(JRvN\ W N)' The question how to define the domain V(1iM) C L1 (JRvN x (JRvN\ W N)) was discussed in the important paper [Kotl] (see Appendix I). Formally, the operator 1iN at t > 0 acts as the pseudo-Liouville operator [BB][DE][EDHL] N (1iNIN)(X) = ({fN,HN})(x)+a 2 L.

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